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WP-AD 2011-14
On downside risk predictability through
liquidity and trading activity:
a quantile regression approach
Antonio Rubia and Lidia Sanchis-Marco
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WP-AD 2011-14
On downside risk predictability through
liquidity and trading activity:
a quantile regression approach
Antonio Rubia and Lidia Sanchis-Marco *
Abstract
Most downside risk models implicitly assume that returns are a sufficient statistic with which to
forecast the daily conditional distribution of a portfolio. In this paper, we address this question
empirically and analyze if the variables that proxy for market liquidity and trading conditions
convey valid information to forecast the quantiles of the conditional distribution of several
representative market portfolios. Using quantile regression techniques, we report evidence of
predictability that can be exploited to improve Value at Risk forecasts. Including trading- and
spread-related variables improves considerably the forecasting performance.
Keywords: Value at Risk, Basel, Liquidity, Trading Activity.
* We would like to thank David Abad, Jos van Bommel, Lucía Cuadro, Christian Riis Flor, Angel León,
Antonio Machado, Simone Manganelli, Manuel Moreno, Alfonso Novales, Alberto J. Plazzi, Gonzalo
Rubio, Allan Timmerman, Ross Valkanov, participants in XVII Finance Forum (Madrid, Spain), EFMA
2010 Meeting (Aarhus, Denmark), EAA 2010 Meeting (Istanbul, Turkey), 2010 ASEPELT Meeting
(Alicante, Spain) and seminar attendants at University of Valencia for their valuable comments and
suggestions to previous versions of this paper. Any error is our own responsibility. Financial support from
the Spanish Department of Science and Innovation through ECO2008-02599 project and Ivie is gratefully
acknowledged. A. Rubia: University of Alicante. L. Sanchis-Marco: University of Castilla-La Mancha.
Corresponding author: lidia.sanchis@uclm.es.

1 Introduction
Implementing risk control and monitoring systems requires quantitative procedures
to capture the level of underlying uncertainty and make accurate predictions. The
Basel Committee on Banking Supervision (BCBS) has contributed greatly to the
popularization of certain international standards, known as Basel I and II Accords,
in the …nancial services industry. This regulatory setting entitles eligible …nancial
institutions to use internal models based on the Value-at-Risk (VaR) measure for
meeting market risk capital requirements. It is agreed that, without the e¤orts
made to comply with the BCBS standards, the …nancial industry would likely be
facing a deeper crisis. The depth of the economic turmoil has shown the necessity
for new standards to achieve greater banking sector resilience and the need to
improve the existing procedures for quantifying market risk. 1 The present paper
is motivated by this concern.
The existing literature has suggested a number of procedures for forecasting
downside risk, mainly VaR, which largely di¤er in their degree of sophistication:
From the simple EWMA approach popularized by RiskMetrics to the more advanced
probabilistic settings based on the Extreme Value Theory; see Manganelli
and Engle (2004) and McNeil et al. (2005) for a review. Remarkably, a number
of empirical studies have revealed that most of these methods do not seem to perform
successfully in practice under standard backtesting techniques (e.g., Kuester
et al., 2006), which underlines the practical complexity that lies behind the simple
notion of VaR. Why is accurate VaR forecasting so elusive? Whereas most
of the previous literature has attempted to address this question on the grounds
of model misspeci…cation, in this paper we adopt an alternative view within the
framework of model risk and analyze the important role played by the set of conditioning
information. In spite of the large methodological di¤erences, the existing
methodologies to model market risk share a common characteristic: They all rely
exclusively on the information conveyed by historical returns. Naturally, this may
turn out to be unnecessarily restrictive for practical purposes, since the conditional
loss function of a portfolio may exhibit non-trivial links with the state variables
that characterize the market environment and trading conditions and which may
help forecasting bursts in volatility and illiquidity, particularly, in times of stress. 2
1 The BCBS is currently carrying out a reform program that aims to enhance Basel II towards
a new regulatory setting on the basis of higher supervisory standards (Basel III), and which is expected
to be completed by the end of 2010. The BCBS has issued several consultative documents
in the interim outlining the main features of the new regulatory setting. It is interesting to note
that, despite its limitations, the VaR paradigm is still a pillar in the internal model approach for
market risk adopted in Basel III, although further re…nements, such as stressed VaR and stress
tests, will also be required to provide a complementary risk perspective.
2 The implicit belief that returns subsume all the relevant information to forecast downside
risk may be originated in a conservative interpretation of the E¢ cient Market Hypothesis. This
1
3

In this paper, we address empirically the premise that certain state variables
that are observable in the market trading process exhibit predictive power to forecast
the tail of the conditional loss distribution and, consequently, are useful for
risk management. Although predictability is not necessarily limited to the set of
variables we analyze, our main focus is on bid-ask spread and volume-related measures.
Our study is motivated by previous …ndings and theoretical considerations
in the asset pricing and market microstructure literature which have underlined
the link between returns and market liquidity, activity, and private information
arrivals. Like returns, liquidity- and volume-related variables are available on the
trading-basis and are highly sensitive to information ‡ow. Like volatility, these
variables are believed to re‡ect collective expectations, environmental conditions
and market sentiments which have a major in‡uence on investor decisions. In
contrast to returns and volatility, however, trade-related variables seem to have
been ignored in the literature devoted to downside risk modelling, even though
there exists previous evidence that supports the predictive power of volume and
liquidity variables on volatility (Suominen, 2001; Bollerslev and Melvin, 1994).
The main aim of this paper, therefore, is to analyze empirically whether downside
risk forecasts can be enhanced by using this information or if, on the contrary,
the predictability of the conditional loss distribution is limited to past returns
and their volatility, as implicitly assumed in most of the empirical models used in
practice.
More speci…cally, we study the performance of several liquidity- and volumerelated
predictors in day-ahead VaR forecasting at di¤erent quantiles using daily
log returns from volume- and value-weighted market portfolios, book-to-market
(B/M), and size-sorted portfolios from the US Stock Exchange in the period
01/1988 through 12/2002. 3 As in the literature concerned with return predictability,
the simplest way to appraise forecastability is to use simple least-squares analysis
in predictive linear regressions; see, for instance, Cochrane (2005). However,
conditional quantiles are unobservable and have to be estimated or, at best, modelled
as a latent process, which makes such an approach infeasible. Fortunately, the
Quantile Regression theory (Koenker and Bassett, 1978) allows us to analyze formally
predictability in the conditional percentiles without departing signi…cantly
from the intuitive spirit that characterizes predictive regressions. Using quantile
regression we can directly model the tail of the conditional distribution of returns
through a functional form that relates the VaR time-varying dynamics to its own
forbids the systematic predictability of returns on the basis of the available information, i.e.,
posits an orthogonal condition on the …rst-order conditional moment. However, it remains silent
about higher-order moments, such as conditional volatility, or other distributional features of
returns, such as conditional percentiles. Furthermore, …nancial markets largely depart in practice
from the complete-market and symmetric-information hypotheses that underlie a number of
theoretical models in the asset pricing literature. In the presence of asymmetric or imperfect
information and other frictions, even the conditional mean of returns may be predictable.
3 Three main reasons prompted us to consider this speci…c sample: i) market-portfolio data
allow us to eliminate the idiosyncratic noise that may a¤ect the main conclusions on individual
stocks, ii) the sample period is particularly interesting for risk management as it includes a stress
scenario originating in the burst of the technological bubble in 2000, and iii) we can analyze the
aggregate measures of liquidity and volume that have been used previously in several studies;
see Chordia et al. (2001). We thank Prof. A. Subrahmanyam for making these data available.
2
4

past as well as to lagged predictors without making an explicit assumption on
the distribution of returns, building on the so-called CAViaR model setting in
Engle and Manganelli (2004). A restricted version of this model, which considers
returns-related information solely, can be taken as a natural benchmark to address
statistically the incremental value of liquidity- and volume-related variables.
Even more importantly, these models can be used to construct VaR forecasts,
which allows us to resort to standard backtesting and statistical techniques (e.g.,
Christo¤ersen, 1998; Piazza et al., 2009) to analyze the actual out-of-sample performance.
Our main empirical conclusions largely support the suitability of the
liquidity- and volume-related variables in forecasting daily VaR.
This paper belongs to the stream of literature that has focused on quantile
regressions for VaR modelling and forecasting; see, among others, Taylor (1999,
2000, 2008), Kouretas and Zarangas (2005), Bao et al. (2006) and Kuester et
al. (2006). The distinctive feature of our study is the analysis of the predictive
ability of the observable variables that the proxy for liquidity and trading-activity
in day-ahead VaR modelling. To the best of our knowledge, no previous paper
has focused on this important aspect, although some recent studies can be related
to this analysis. Cenesizoglu and Timmerman (2008) analyze the predictability of
the distribution of monthly returns on a set of state variables that are believed to
predict the equity premium, such as valuation and corporate ratios, bond yields,
and di¤erent measures of default and market risk. Some of these variables are
shown to be helpful in predicting di¤erent quantiles, particularly, at the right
tail of the distribution, which may lead to more e¢ cient portfolio selection and
option trading. Adrian and Brunnermeir (2009) model weekly VaR dynamics in
the banking industry using a similar set of state variables aiming to characterize
the determinants of CoVaR dynamics. Our paper contributes to this literature by
providing novel evidence on i) the predictability of the left tail of the conditional
distribution of daily returns on the basis of volume and liquidity measures, and ii)
the suitability of these variables for downside risk forecasting. This study is also
related to the previous studies focused on the analysis of return predictability and
the links between volatility and the main variables related to the trading ‡ow, such
as bid-ask spreads and trading-based measures; see, among others, Clark (1973),
Tauchen and Pitts (1983), Stoll (1989) and Kalimipalli and Warga (2002).
The remaining part of the paper is organized as follows. Section 2 reviews the
basic elements in VaR modelling and brie‡y introduces the quantile regression approach.
Section 3 describes the main features in the data set analyzed and carries
out the empirical analysis. The main analysis focuses on the volume-weighted market
portfolio, using both (quantile) predictive regressions and backtesting analysis
techniques. For the sake of completeness, we also analyze predictability for di¤erent
characteristic portfolios, namely, value-weighted, book-to-market (B/M) and
size-sorted market portfolios. Finally, Section 4 summarizes and concludes.
3
5

2 Modelling and forecasting downside risk: Value
at Risk
Let r t ; t = 1; :::; T , be the daily log-return time-series of a portfolio, and let F t
be the natural …ltration that includes all the available information at time t; such
as any measurable transformation on the past observations of r t as well as any
other observable variable. For simplicity, but no loss of generality, we assume in
the sequel that returns behave as a stationary martingale di¤erence sequence such
that E (r t jF t 1 ) = 0; with bounded moments E jr t j < 1 for some > 2 large
enough. 4 For a probability 2 (0; 1), we de…ne the (1 )% VaR of a …nancial
asset or portfolio as the maximum loss over a horizon of h days which is expected
at the (1 )% con…dence level given F t , i.e., the -quantile of the conditional
loss distribution of the portfolio. Formally, we can denote:
V aR ;t+h = finf x 2 R : Pr (r t;h xjF t ) g (1)
= fQ ;t (h)g
with Q ;t (h) de…ned implicitly, and r t;h = P j=1;h r t+j denoting the h-period return.
In market risk management, h typically ranges from 1 to 10 days and =
f0:01; 0:05g. For instance, the Basel market risk framework stresses the use of the
1% conditional percentile to determine capital adequacy, while traders in a bank
are often constrained by the rule that the 95% VaR of their position should not
exceed a given bound on a daily basis (McNeil et al., 2005). The extant literature
in …nancial econometrics has suggested very di¤erent procedures with which to
model and forecast VaR dynamics. In the following subsection, we discuss the
main characteristics of the quantile regression approach. Appendix A sketches the
main features of several alternative procedures that shall be used together with
quantile regression in the empirical analysis in Section 4.
2.1 Quantile regression: CAViaR models
Following Koenker and Bassett (1978) and Basset and Koenker (1982), the expected
conditional quantile could be written as a F t -measurable linear function
of n explanatory variables, X t = (x 1t ; :::; x nt ) 0 ; and an (n 1) vector of unknown
coe¢ cients that depends on the -quantile, namely, Q ;t (h ) = X 0 t . Particularizing
for a one-day holding period, h = 1, this formulation is equivalent to
assume the quantile regression model
r t+1 = X 0 t + u t+1; ; t = 1; :::; T (2)
4 This condition is not strictly necessary but simpli…es exposition considerably. It comes with
no loss of generality because it is customary to demean (daily) returns previously to ensure that
the resultant series behaves as a martingale di¤erence sequence; see, for instance, Taylor (2008).
In Section 4, we …lter out the predictable pattern in the conditional mean of the di¤erent portfolio
returns by …tting an AR(1) model prior to apply the VaR methods.
4
6

where u t; is an error term satisfying E (u t; jX t 1 ) = 0: As in the standard regression
model, this general speci…cation does not impose any particular restriction on
the distribution of the data.
Model (2) is highly reminiscent of the standard linear regression setting. Whereas
the least-squares analysis attempts to characterize the conditional mean of the distribution
as a function of the regressors, E (r t+1 jX t ) ; the quantile regression allows
us to model directly the dynamics of the conditional -quantile of the distribution.
A well-known particular case arises for = 1=2 (also known as least absolute
deviation regression model), which is intended to provide estimates of the slope
coe¢ cients for the conditional median of the process. In this case, the relevant
coe¢ cients can be estimated consistently by minimizing the sum of the absolute
values of the residuals. More generally, given an arbitrary value of 2 (0; 1) ; the
unknown vector of parameters in (2) can be estimated consistently as:
( TX
b : arg min ju t+1; j I (rt+1
2R n Xt 0 ) +
t=1
)
TX
(1 ) ju t+1; j I (rt+1

that in the Xt
variable. The main purpose of the autoregressive structure is to
ensure that the dynamics of the conditional quantile change smoothly over time.
Since VaR dynamics are highly persistent, the lag of the dependent process could
also be seen as an instrumental variable that proxies for the true latent process.
Similarly, the variable jr t 1 j is a natural proxy for the unobservable volatility of
returns. Since it introduces a source of (stochastic) short-term variation related to
the arrival of news in the market, this process is expected to be a major driver in
any market risk measure. At this point, the similarities between the basic structure
of the CAViaR model under the restriction = 0 (so-called Symmetric Absolute
Value CAViaR model, Restricted-CAViaR henceforth) and the class of GARCH
models widely used to characterize volatility are fully evident.
In addition to the GARCH-type architecture, the VaR dynamics in the CAViaR
setting could be driven by the lagged values of a set of state variables. The existing
literature has not yet discussed which variables should be included in such an
analysis. For daily market data, the most obvious candidates seem to be the
variables that proxy for liquidity and other trade conditions in the market. The
main problem in this regard is that market liquidity is a di¢ cult magnitude to
measure and one which is not easy to relate to a single aspect of the market. The
central strategy we adopt consists of individually analyzing a number of tradebased
and order-based market variables (see Section 3.1 for details) which are
widely accepted to be related to liquidity in order to detect predictability. Note
that, although the results in a parametric modelling may be sensitive to the choice
of a particular proxy of liquidity or another, we should expect a robust picture to
emerge when considering a wide range of proxying variables. The model resulting
from extending the basic Restricted-CAViaR speci…cation with the lags of a single
predictor can be seen as a low-order individual autoregressive distributed lag model
for the conditional quantile. This class of models is known to be particularly useful
in the forecasting analysis; see, for instance, Rapach and Strauss (2009), and allows
us to examine how a model extended with a variety of individual proxies of liquidity
performs relative to the restricted model.
Finally, under suitable regularity conditions and as the sample size is allowed
to grow unbounded, it can be shown (cf. Engle and Manganelli 2004, Thms. 1
and 2) that:
p
T ( b ) ! d N (0; V ) (5)
i.e., b is a p T -consistent estimate of the unknown vector , and the (suitably rescaled)
estimation bias is asymptotically distributed as a normal distribution with
zero mean and …nite covariance matrix V . In order to consistently estimate this
matrix, Engle and Manganelli (2004) propose a robust estimator that combines
kernel density estimation with the heteroskedasticity-consistent covariance matrix
estimator of White (1980); see Engle and Manganelli (2004, Thm. 3) for details.
We shall use this approach to perform formal inference on the signi…cance of the
estimate.
6
8

3 Empirical analysis
3.1 Data
We consider the returns of di¤erent representative market portfolios in order to
characterize the VaR dynamics. In particular, our basic data set comprises continuously
compounded returns from the volume- and value-weighted portfolios in the
US market over the period 01-04-1988 to 12-31-2002, totaling 3,782 daily observations.
These data are available from CRSP and shall be used in our main analysis
in Section 4.2. In addition, we analyze daily log returns on the B/M-sorted and
size-sorted portfolios in Section 4.3. These data are available at Kenneth French’s
website.
Along with daily portfolio log returns, we observe daily data of several potentially
predictive variables throughout the period analyzed and which are constructed
from individual …rm bid-ask spreads and volume data; see Chordia et al.
(2001) for details. These variables can be classi…ed as:
i) Trading-related variables: Trading Volume (V) measured in thousands of shares;
Number of Trades (NT) calculated as the sum of sell and buy trades; Number of
Sell trades (NS); Number of Shares Sold in thousands (NSS) and Traded Volume
in Dollars (TVD).
ii) Liquidity variables: Quoted Spread (QS) measured as the dollar di¤erence
between ask and bid prices; E¤ective Spread (ES) given by the signed di¤erence
between trade price and bid-ask midpoint (MP); Relative Quoted Spread (RQS)
de…ned as the ratio QS/MP, and Relative E¤ective Spread (RES) de…ned as the
ratio ES/MP. 7
[Insert Table 1 around here]
Table 1 displays in Panel A the usual descriptive statistics for the demeaned
time-series of the di¤erent portfolio returns analyzed in the paper, and all the
explanatory variables (in logarithms) used in our analysis in Panel B. Returns exhibit
the characteristic stylized features in daily samples: Excess of kurtosis, mild
degree of skewness and negligible autocorrelation. The most salient feature of the
predictors in the trade-based and the order-based measures is the strong degree of
persistence as measured by the …rst-order autocorrelation coe¢ cient. Returns are
contemporaneously correlated to all the variables analyzed (correlations are not
shown for restricteding space but are available upon request). In particular, returns
are positively correlated with the variables in the volume group (the average
correlations are around 39%) and negatively correlated to liquidity-related variables
(the average correlations are around -25%). As usual, the variables within
each group are strongly correlated among themselves, and largely and negatively
correlated with the variables in the other group. Cross-correlations range from
-79%, for TVD and QS, to -88%, for TVD and QS.
7 In addition to these variables, we considered alternative variables which did not led to qualitative
changes over those reported in the next section. For instance, considering the logarithm
transformation of the volume or the unexpected volume –measured as the residuals from an
AR(1) model–does not seem to have a major change in the out-of-sample ability of the model.
These results are not presented to save space but are available from the authors upon request.
7
9

Tables
Table 1: Descriptive Statistics.
Panel A: Returns
r t Mean Median Max. Min. Var Skew. Kurt. (1)
r t;V ol 0.00 0.01 5.54 -6.70 0.98 -0.20 7.78 -0.06
r t;V alue 0.02 0.06 4.97 -6.11 0.49 -0.35 10.84 -0.07
r 1t , B=M 0.01 0.02 6.65 -7.85 1.22 -0.11 7.29 -0.07
r 2t , B=M 0.00 0.02 4.81 -6.32 0.67 -0.43 8.36 -0.05
r 1t;size 0.00 0.05 6.02 -7.56 0.66 -0.71 11.06 -0.05
r 2t;size 0.00 0.01 5.73 -6.89 1.05 -0.16 7.39 -0.06
Panel B: Predictive variables
Xt Mean Median Max. Min. Var Skew. Kurt. (1)
V 7.39 7.21 9.69 5.52 0.73 0.36 1.86 0.96
NT 6.85 6.69 8.76 5.07 0.65 0.30 1.64 0.98
NS 6.09 5.94 8.02 4.24 0.65 0.29 1.67 0.98
NSS 6.51 6.33 8.80 4.50 0.73 0.35 1.86 0.96
TVD 11.10 11.04 13.00 9.13 0.76 0.17 1.66 0.96
QS -1.89 -1.74 -1.20 -3.40 0.21 -1.64 4.79 0.99
ES -2.29 -2.11 -1.50 -3.80 0.20 -1.53 4.48 0.99
RQS -5.39 -5.39 -4.80 -6.90 0.20 -1.02 3.23 0.99
RES -5.96 -5.76 -5.20 -7.20 0.20 -0.96 3.08 0.99
This table shows in Panel A the descriptive statistics (mean, median, maximum, minimum,
variance, skewness and kurtosis) of the demeaned returns (previously multiplied by
100) for the volume and value weighted market portfolio, B/M and Size sorted portfolios
corresponding to Low30 (r 1t
) and High30 (r 2t
). Panel B shows the descriptive analysis
for the explanatory variables involved in the analysis (in logarithms). The last column
indicates the …rst-order autocorrelation of the variables. The variables included are V
(trading Volume); NT (Number of Trades); NS (Number of Sell trades); NSS (Number of
Shares Sold in thousands); TVD (Traded Volume in Dollars); QS (Quoted Spread); ES
(E¤ective Spread); RQS (Relative Quoted Spread) and RES (Relative E¤ective Spread).
25
10

3.2 Market portfolio analysis
Throughout the following subsections we analyze the ability of the predictive variables
to forecast day-ahead VaR for the returns of the volume-weighted market
portfolio, both through a predictive regression that involves the whole sample
and an out-of-sample predictive analysis in a rolling-window approach. The focus
on day-ahead estimation is consistent with the holding period considered for
internal risk control by most …nancial …rms (Taylor, 2008). Owing to their empirical
relevance in risk management, we are particularly interested in the quantiles
2 f0:01; 0:05g ; but shall also analyze a wider range of probabilities given by
= f0:01; 0:025; 0:05; 0:075g to characterize predictability along the left tail of
the distribution. It should be mentioned that in this section we also considered the
analysis on the returns of the value-weighted market portfolio, which are not presented
for saving space as the qualitative evidence is completely analogous to that
discussed below. The complete analysis is available from authors upon request.
The daily returns of the volume-weighted market portfolio over the total period
analyzed (see Figure 1 below) include di¤erent episodes in terms of activity and
volatility. The beginning of the sample corresponds to the period that followed the
market crash in October 1987. After the extraordinary crash, the volatility of the
market decreased progressively and reverted to normal levels. In 1998, Long-Term
Capital Management failure in the hedge-fund industry led to a massive bailout
by other major banks and investment houses that, in turn, generated an excess
of volatility in the market and which preceded the burst of the dot-com …rms
in 2000. Finally, the data from 2000 onwards show the large excess of volatility
that characterized the market after the burst of the technological bubble. This
particular period, depicted by the grey line in Figure 1, will be analyzed in detail
in the out-of-sample analysis (back test) later on.
[Insert Figure 1 around here]
3.2.1 Predictive regression analysis
We …rst analyze predictability through predictive-like quantile regressions. More
speci…cally, we estimate the unrestricted CAViaR model
V aR ;t = ;0 + ;1 V aR ;t 1 + ;2 jr t 1 j + j log (X t 1 ) j (6)
for any of the conditioning variables X t described in Section 3.1, using the entire
sample, from 01-04-1988 to 12-31-2002, with 3; 782 observations. Our main
interest is to test the suitability of these unrestricted models against a restricted
CAViaR speci…cations based solely on returns. Clearly, a rejection of the null hypothesis
H 0 : = 0 for a speci…c model and quantile provides formal evidence
on the suitability of the posited variable as a predictor of the conditional distribution.
The objective function in the quantile regression minimization problem is
optimized through the Simulated Annealing optimization algorithm (Go¤e, Ferier
and Rogers, 1994), while the asymptotic covariance matrix is estimated using a
robust sandwich-type estimator based on a k-nearest neighbor kernel, as described
8
11

Figures
6
4
2
Market Portfolio Returns
0
­2
­4
­6
Out of Sample Period
­8
1988/01/04 1991/10/01 1995/06/27 1999/03/25 2002/12/31
Observations(Data)
Figure 1: Returns of the volume weighted market portfolio.
3.5
3
Restricted
Volume
RQS
ForecastsVaR (5% nominal level)
2.5
2
1.5
1
1998/09/10 1999/09/10 2000/09/10 2001/09/10
Observations (Data)
Figure 2: Forecasts VaR at nominal level 5% of Restricted-CAViaR
model (solid line) and Unrestricted CAViaR model extended with
volume and Relative Quoted Spread (RQS) variable (grey and black
dashed lines respectively).
34
12

in Engle and Manganelli (2004). 8
[Insert Table 2 around here]
Table 2 reports the estimated coe¢ cients and the one-sided robust p-values for
the all the variables for 2 f0:01; 0:05g. Following Engle and Manganelli (2004),
we set the bandwidth-type parameter in the robust estimation of the covariance
matrix to k = 40 and k = 60 for the 1% and 5% quantiles, respectively. Several
features are worth commenting upon. Firstly, the resulting estimates show the
strong degree of persistence in the VaR dynamics as measured by the estimate of
the autoregressive coe¢ cient, b ;1 . The degree of persistence is weaker for =
0:01 because extreme percentiles are more likely to be driven by jumps (outlying
observations) in data generating process of returns, which are expected to exhibit
a more random, irregular behavior. Secondly, and accordingly with the previous
result, the average in‡uence of the volatility process on the VaR estimates, as
measured by the coe¢ cient b ;2 ; becomes more important as the size of the quantile
decreases. These results completely agree with the qualitative evidence discussed,
among others, in Engle and Manganelli (2004). Turning our attention to the
empirical relevance of the di¤erent predictors, all the variables analyzed have a
positive e¤ect on the conditional distribution of returns. Increments in the volumeor
spread-related variables tend to generate larger levels of VaR, as might be
expected from the theoretical considerations and previous evidence discussed in
Section 1. The analysis on the signi…cance of the estimated coe¢ cients o¤ers,
however, mixed results. Whereas the null hypothesis of no predictability is strongly
rejected for any of the variables analyzed for = 0:05, it cannot be rejected at
any of the usual con…dence levels for the 1% percentile.
There are two possible explanations for this …nding. On the one hand, it is possible
that a variable has predictive power at a given probability yet not at another.
Thus, the statistical analysis would indicate a heterogeneous ability in market liquidity
to predict the tail of the return distribution, with extreme percentiles being
harder to foresee. Since the outlying observations that characterize the left tail
of the returns distribution are related to the jump-component of its data generating
process, and since this component is likely to be driven by a non-regular
process, this point seems plausible: outliers and other extreme movements may
be too volatile and irregular to be predicted systematically by covariates other
than market volatility itself. On the other hand, the discrepancies may be the
consequence of statistical distortions, particularly, for small percentiles such as
= 0:01. In fact, there are two major sources of potential biases in this context:
(small-sample) biases stemming from the robust estimation of the covariance matrix
V , and biases resulting from dealing with a target probability which is close
to the boundary.
8 Simulated annealing is a local random-search search algorithm that can accept values that
increase the objective function (rather than lower it) with a probability that decreases as the
number of iterations increases. The main purpose is to prevent the search process from becoming
trapped in local optima, which in addition provides low sensitivity to the choice of the initial
values. To minimize the possibility of getting convergence to local optima, the optimization
process was repeated 1; 000 times over the whole sample.
9
13

Table 2: Inference results from predictive quantile regressions.
X t
= 5% = 1%
^ ;0
^;1 ^;2 ^
^ ;0
^;1 ^;2 ^
V -0.04 0.96 0.05 0.01 0.02 0.92 0.11 0.01
(0.00) (0.00) (0.00) (0.00) (0.37) (0.00) (0.06) (0.16)
NT -0.03 0.96 0.05 0.01 0.03 0.92 0.11 0.01
(0.00) (0.00) (0.00) (0.00) (0.25) (0.00) (0.04) (0.13)
NS -0.02 0.96 0.05 0.01 0.03 0.92 0.11 0.01
(0.00) (0.00) (0.00) (0.00) (0.24) (0.00) (0.04) (0.16)
NSS -0.03 0.96 0.05 0.01 0.03 0.92 0.11 0.01
(0.00) (0.00) (0.00) (0.00) (0.27) (0.00) (0.06) (0.19)
TVD -0.06 0.96 0.05 0.01 -0.01 0.92 0.12 0.01
(0.00) (0.00) (0.00) (0.00) (0.41) (0.00) (0.05) (0.12)
QS -0.01 0.97 0.05 0.01 0.07 0.92 0.14 0.01
(0.01) (0.00) (0.00) (0.00) (0.12) (0.00) (0.08) (0.32)
ES -0.00 0.97 0.05 0.00 0.08 0.91 0.15 0.01
(0.40) (0.00) (0.00) (0.01) (0.14) (0.00) (0.11) (0.31)
RQS -0.04 0.97 0.05 0.01 0.04 0.92 0.14 0.01
(0.01) (0.00) (0.00) (0.00) (0.32) (0.00) (0.08) (0.36)
RES -0.08 0.96 0.05 0.02 -0.03 0.89 0.15 0.03
(0.05) (0.00) (0.00) (0.05) (0.34) (0.00) (0.13) (0.18)
This table shows the estimated parameters and robust p-values (in brackets) for the
entire sample and the quantile regression model (4),
V aR ;t = ;0 + ;1 V aR t 1 + ;2 jr t 1 j + j log(X t 1)j);
given = 0:05 and = 0:01: The …rst column shows the volume-related and liquidity
variables Xt
analyzed.
26
14

The theoretical arguments that support consistency in the robust estimation of
V hold asymptotically. As with other robust estimates of the covariance matrix,
the correct choice of the bandwidth-type parameter in the kernel-type estimation
involved plays a critical role, particularly, in …nite samples. Furthermore, when
is close to the boundary limits, the asymptotic variance increases because of
the small density of the observations in that region. The conjunction of these
factors may cause signi…cant biases in the estimates of the covariance matrix and
meaningful distortions in the subsequent hypotheses testing. In order to analyze
the sensitivity of the results to the estimation of V , we estimated the CAViaR
models for 2 with the bandwidth-type parameter taking values over a wide
range, k 2 f10; 30; 50; 70; 90g. Table 3 summarizes the main results from this
analysis, showing the estimates of the and ;2 coe¢ cients and their robust
p-values given k: For the remaining parameters, there was no qualitative changes
and, therefore, we do not present results in order to save space, although these are
available upon request. Table 3 reveals several interesting …ndings. For relatively
large values of in the tail of the distribution, the empirical evidence strongly
supports predictability for all the variables, independently of the value of k; which
allows us to conclude that predictable relationships do exist. Nevertheless, as
decreases towards the zero limit, the statistical evidence of predictability weakens
and eventually vanishes. Overall, the evidence of predictability seems to be more
evident for the variables in the volume-related group, in which the null H 0 : = 0
is rejected for most variables, and (marginally) rejected even for = 0:01. The
overall analysis also reveals that the main statistical conclusions in this analysis do
not seem to be particularly sensitive to the choice of the bandwidth-type parameter
k.
[Insert Table 3 around here]
The second potential source of biases in this context is related to the existence
of statistical distortions in the quantile regression methodology when dealing with
quantiles close to the boundaries. This well-known problem originates from the
fact that conventional large sample theory does not apply su¢ ciently far in the
tails, a behavior which worsens in …nite samples and which may lead to unsound
inference; see, for instance, Chernozhukov (2005) for a detailed discussion. As
a result, inference may be misleading. Note, for instance, that the signi…cance
analysis of b ;2 in Table 3 leads to puzzling conclusions. Whereas the estimates
of the coe¢ cient related to market volatility largely increases as is smaller,
standard errors increase reducing the degree of statistical signi…cance and causing
H 0 : ;2 = 0 not to be rejected for some nominal levels. This suggests that VaR
dynamics may not be driven by volatility, precisely in the context values in which
total market volatility is the most likely driver of process. This counterintuitive
feature seems a statistical artifact rather than a true rejection of signi…cance.
Owing to the statistical di¢ culties in dealing with extremes percentiles, empirical
studies often avoid the analysis for values of close to boundaries; see, for
instance, Cenesizoglu and Timmerman (2008). In our context, however, we have
alternative methods with which to study the forecasting ability of the predictive
variables even for such percentiles. Since the ultimate purpose of VaR modelling is
to generate accurate market risk forecasts, the most important question is whether
10
15

Table 3: Sensitivity analysis of p-values to di¤erent k-bandwidth.
V NT NS NSS TVD QS ES RQS RES jr t 1 j
7.5% ^ 0.01 0.01 0.01 0.01 0.01 0.02 0.00 0.01 0.02 0.06
10 (0.00) (0.00) (0.00) (0.00) (0.24) (0.00) (0.02) (0.00) (0.13) (0.00)
30 (0.00) (0.01) (0.00) (0.00) (0.00) (0.00) (0.04) (0.07) (0.02) (0.00)
k 50 (0.00) (0.01) (0.00) (0.00) (0.00) (0.00) (0.02) (0.06) (0.06) (0.00)
70 (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.03) (0.07) (0.04) (0.00)
90 (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.02) (0.04) (0.02) (0.00)
5.0% ^ 0.01 0.01 0.01 0.01 0.01 0.01 0.00 0.01 0.02 0.05
10 (0.02) (0.00) (0.01) (0.03) (0.00) (0.00) (0.00) (0.00) (0.16) (0.00)
30 (0.00) (0.01) (0.00) (0.00) (0.00) (0.04) (0.00) (0.00) (0.06) (0.00)
k 50 (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.01) (0.00) (0.04) (0.00)
70 (0.00) (0.00) (0.00) (0.01) (0.00) (0.00) (0.02) (0.00) (0.07) (0.00)
90 (0.01) (0.00) (0.01) (0.01) (0.01) (0.01) (0.00) (0.01) (0.05) (0.00)
2.5% ^ 0.02 0.02 0.02 0.03 0.02 0.01 0.00 0.00 0.03 0.07
10 (0.04) (0.14) (0.43) (0.00) (0.00) (0.04) (0.34) (0.39) (0.46) (0.01)
30 (0.03) (0.17) (0.10) (0.04) (0.07) (0.10) (0.29) (0.36) (0.08) (0.11)
k 50 (0.02) (0.08) (0.06) (0.02) (0.04) (0.12) (0.26) (0.35) (0.11) (0.09)
70 (0.03) (0.03) (0.04) (0.02) (0.02) (0.11) (0.27) (0.35) (0.06) (0.07)
90 (0.03) (0.03) (0.03) (0.02) (0.02) (0.07) (0.28) (0.35) (0.04) (0.07)
1.0% ^ 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.03 0.14
10 (0.04) (0.02) (0.05) (0.11) (0.03) (0.24) (0.03) (0.30) (0.00) (0.03)
30 (0.15) (0.16) (0.12) (0.15) (0.14) (0.34) (0.33) (0.38) (0.24) (0.05)
k 50 (0.12) (0.12) (0.11) (0.14) (0.11) (0.33) (0.28) (0.38) (0.15) (0.03)
70 (0.14) (0.15) (0.14) (0.16) (0.10) (0.28) (0.24) (0.30) (0.16) (0.03)
90 (0.11) (0.13) (0.11) (0.12) (0.10) (0.28) (0.24) (0.32) (0.15) (0.03)
This table shows the estimated coe¢ cients ^ ;2 and ^ and robust p-values (in brackets
and in bold) of the test for individual signi…cance from model (4) and the entire sample
when the robust asymptotic covariance matrix is estimated with a kernel with values of
k 2 f10; 30; 50; 70; 90g in the covariance matrix estimation process for a larger set
of quantiles. The ^ ;2 estimates (last column) are from model (4) with Xt = V:
27
16

the covariate-extended risk models have real out-of-sample predictability or not.
Therefore, the backtesting analysis provides us with a more appropriate framework
with which to compare the relative performance of the models, and to attempt to
disentangle the predictive ability of the model for any quantile. This issue is
studied carefully in the following subsection.
3.2.2 Out-of-sample analysis: backtesting analysis
In this section we analyze the day-ahead out-of-sample performance of the covariateextended
CAViaR models relative to the restricted Restricted-CAViaR for 2 :
In addition, we also analyze the relative performance of the covariate-extended
model against several established methods that are based solely on returns, such
as the EWMA model (RiskMetrics), the Gaussian GARCH(1,1) model, and a
model that combines GARCH estimation with a block-maxima approach in the
Extreme Value Theory. The main characteristics of these alternative procedures
are sketched in Appendix A; see McNeil et al. (2005) for further details.
Following Engle and Manganelli (2004) and Alexander and Sheedy (2008) we
consider a rolling-window estimation period formed with the most recent 2; 700
observations available at any day to initialize the risk models and generate VaR
forecasts for the probabilities 2 . This allows us to construct a sequence
of N = 1; 000 one-day ahead daily VaR predictions that can be compared with
realized returns. 9 Whereas backtesting is a crucial element in the validation process
of an internal risk model, a de…nitive methodology has yet to be determined. In
the absence of a regulatory setting, the existing literature has adopted certain
statistical standards that we brie‡y discuss in the sequel. Appendix B provides a
more detailed discussion.
The most popular backtesting procedures analyze the statistical properties of
an indicator variable (known as exception or VaR violation) that signals the occurrence
of a realized return exceeding the expected VaR level, i.e., a random
process, here denoted as H ;t , that takes value one if r t < V aR ; and zero otherwise,
t = 1; :::; N. A desirable property is that the realized number of exceptions
should represent approximately a % of the total out-of-sample period. This property
is known in risk management as reliability and suggests the testable condition
H 0;U : E (H ;t ) = : Also, the dynamic properties of H ;t should resemble those of
an independent and identically distributed process, which combined with the reliability
property renders the joint property of perfect conditional coverage, namely,
H 0;C : E (H ;t jF t 1 ) = .
Christo¤ersen (1988) proposed a sequence of likelihood-ratio tests to check
the properties of i) correct unconditional coverage, ii) (…rst-order serial) independence,
and iii) correct conditional coverage in the exception variable given i) and
ii); see Appendix B for details. These tests are widely used in practice and, therefore,
constitute our …rst backtesting approach. In the sequel, we shall refer to
these procedures as LR UC , LR IND and LR CC , respectively. Similarly, Engle and
Manganelli (2004) proposed another exception-based testing procedure to test for
9 The average estimates of the parameter estimates over the out-of-sample period show the
same patterns discussed before and, hence, are not presented in order to save space, although
these results are available upon request.
11
17

the crucial H 0;C hypothesis, the so-called dynamic quantile test (henceforth DQ).
This test is widely believed to have better properties than LR CC because it has
power to detect higher-order dependences as it explodes a richer set of information.
Consequently, we shall use the DQ procedure in our analysis. Finally, Piazza et
al. (2009) have recently proposed an encompassing VaR quantile-regression test
intended to test H 0;C : E (H ;t jF t 1 ) = : The distinctive characteristic is that
the test is not de…ned on the exception variable, but on the time-series of predictions
itself. This procedure, denoted as V QR in the sequel, uses a linear quantile
regression to address if the VaR model is correctly speci…ed. It should be noted
that the test is asymptotically equivalent to DQ, yet, as claimed by the authors,
may have better properties in …nite samples. Inference on the basis of quantile
regressions, on the other hand, may su¤er from statistical problems when dealing
with extreme quantiles in …nite samples, as discussed previously.
[Insert Table 4 around here]
We …rst analyze the backtesting results for the risk models based solely on returns:
VaR-GARCH, VaR-EWMA, EVT-BM (extreme value theory) and Restricted-
CAViaR models. Table 4 reports the main outcomes from this analysis, displaying
the empirical frequency of exceptions b H = P N
t=1 H ;t=N; the test statistics of the
LR UC , LR IND , LR CC ; DQ and V QR testing procedures as well as their respective
p-values. For all these models, the empirical unconditional coverage for the percentiles
0:05 tend to be greater than the nominal level, with b H signi…cantly
departing from in most cases. Consequently, the hypotheses of perfect conditional
and unconditional coverages tend to be rejected. Similar evidence has been
reported previously, for instance, in Taylor (2008) for GARCH and CAViaR-type
models; see also Piazza et al. (2009).
For quantiles < 0:05; the distortions in the unconditional coverage are considerably
reduced for all but the EWMA model and, therefore, the H 0;U hypothesis
tends not to be rejected. The overall evidence for H 0;C , however, is mixed and depends
on the particular test applied. Whereas the conservative LR CC test accepts
the hypothesis of perfect coverage, the DQ and V QR tests, which are considered
as much more powerful testing procedures, tend to reject the null hypothesis of
correct conditional coverage. Among the di¤erent returns-based VaR procedures
analyzed, the EVT method seems to yield the best performance but, overall, none
of these procedures seems able to pass convincingly the backtesting analysis. Remarkably,
the V QR test rejects the correct performance of all the returns-based
models for any of the conditional quantiles analyzed.
[Insert Tables 5 around here]
Next, we turn our attention to the backtesting results from the volume- and
spread-extended CAViaR models. Table 5 displays the main backtesting results.
The most remarkable feature is that, whereas the standard Restricted-CAViaR
speci…cation and other returns-based approaches exhibit large biases, the inclusion
of the proxies for market liquidity and trading activity conditions improves
the out-of-sample results considerably. The estimated VaR dynamics are shifted
12
18

Table 4: Backtesting VaR analysis for benchmark models. Volume weighted
market portfolio.
MODEL Exc. LR UC LR IND LR CC DQ VQR
EWMA 7.5% 8.9% 2.68(0.10) 0.67(0.41) 3.38(0.18) 23.45(0.00) 14.11(0.00)
5.0% 5.5% 0.51(0.47) 0.00(0.99) 0.52(0.77) 11.40(0.07) 23.54(0.00)
2.5% 1.5% 4.78(0.02) 0.46(0.49) 5.21(0.07) 18.48(0.00) 84.25(0.00)
1.0% 0.5% 3.09(0.08) 0.05(0.82) 3.13(0.21) 3.45(0.74) 177.84(0.00)
GARCH(1,1) 7.5% 11.3% 18.22(0.00) 0.29(0.59) 18.59(0.00) 44.21(0.00) 20.85(0.00)
5.0% 7.4% 10.63(0.00) 0.46(0.49) 11.15(0.00) 29.44(0.00) 31.05(0.00)
2.5% 2.8% 0.35(0.55) 0.08(0.78) 0.44(0.80) 22.51(0.00) 60.93(0.00)
1.0% 0.9% 0.10(0.75) 0.16(0.69) 0.27(0.87) 11.40(0.07) 23.54(0.00)
EVT-BM 7.5% 10.4% 10.85(0.00) 0.58(0.45) 11.49(0.00) 18.24(0.00) 14.91(0.00)
5.0% 6.1% 2.36(0.12) 0.51(0.48) 2.89(0.23) 9.31(0.15) 10.10(0.01)
2.5% 2.4% 0.04(0.83) 0.31(0.58) 0.35(0.84) 3.32(0.76) 46.30(0.00)
1.0% 0.5% 3.10(0.08) 0.04(0.84) 3.13(0.21) 3.64(0.72) 74.36(0.00)
SAV-CAViaR 7.5% 10.1% 8.86(0.00) 0.46(0.49) 8.74(0.01) 28.14(0.00) 9.53(0.01)
5.0% 7.4% 10.63(0.00) 0.50(0.48) 11.19(0.00) 28.91(0.00) 9.63(0.01)
2.5% 3.2% 1.85(0.17) 0.00(0.99) 1.86(0.39) 10.86(0.09) 11.77(0.00)
1.0% 1.3% 0.83(0.36) 0.31(0.57) 1.15(0.56) 14.05(0.02) 27.28(0.00)
This table shows the Backtesting analysis for the one-day forecasts of the VaR given the
EMWA, Gaussian GARCH, EVT and the Restricted-CAViaR (SAV-CAViaR) model
using the volume weighted market portfolio. The second column shows the estimated
ratio of empirical exceptions. LR UC , LR IND , LR CC , DQ and VQR denote the values
of the test statistics for unconditional coverage, independence, conditional coverage, DQ
test and VQR test, respectively, (see Appendix B for details), whereas the p-values of
the respective test statistics are exhibit in brackets.
28
19

Table 5: Backtesting VaR analysis for volume and liquidity extended CAViaR
models. Volume weighted market portfolio. See details in Table 4.
X t Exc. LR UC LR IND LR CC DQ VQR
V 7.5% 8.1% 0.51(0.48) 0.06(0.80) 0.43(0.81) 8.03(0.24) 1.72(0.42)
5.0% 5.5% 0.51(0.47) 0.36(0.55) 0.88(0.64) 7.98(0.23) 1.69(0.43)
2.5% 2.2% 0.38(0.53) 0.94(0.33) 1.32(0.51) 6.02(0.42) 5.36(0.07)
1.0% 0.7% 1.02(0.31) 0.08(0.77) 1.09(0.58) 1.38(0.96) 29.78(0.00)
NT 7.5% 8.1% 0.51(0.48) 0.06(0.80) 0.43(0.81) 7.50(0.27) 1.10(0.58)
5.0% 6.0% 1.98(0.16) 0.61(0.43) 2.61(0.27) 10.92(0.09) 3.15(0.21)
2.5% 2.3% 0.17(0.68) 1.03(0.30) 1.20(0.55) 4.02(0.67) 2.13(0.34)
1.0% 0.9% 0.10(0.75) 0.14(0.70) 0.25(0.88) 2.30(0.88) 33.69(0.00)
NS 7.5% 8.2% 0.69(0.41) 0.03(0.85) 0.55(0.76) 7.06(0.31) 1.42(0.49)
5.0% 5.5% 0.51(0.47) 0.36(0.55) 0.88(0.64) 7.63(0.26) 1.56(0.46)
2.5% 2.0% 1.10(0.29) 0.78(0.38) 1.87(0.39) 5.24(0.51) 5.29(0.07)
1.0% 0.6% 1.89(0.17) 0.06(0.81) 1.94(0.38) 1.99(0.92) 32.95(0.00)
NSS 7.5% 8.0% 0.35(0.55) 0.01(0.91) 0.25(0.88) 6.29(0.39) 1.72(0.42)
5.0% 5.6% 0.73(0.39) 0.28(0.59) 1.03(0.60) 9.53(0.16) 3.39(0.18)
2.5% 1.9% 1.61(0.20) 0.70(0.40) 2.29(0.32) 5.63(0.46) 7.44(0.02)
1.0% 0.6% 1.89(0.17) 0.06(0.81) 1.94(0.38) 2.15(0.90) 53.83(0.00)
TVD 7.5% 8.3% 0.89(0.34) 0.01(0.91) 0.71(0.69) 6.96(0.32) 2.50(0.29)
5.0% 6.1% 2.39(0.12) 1.44(0.23) 3.86(0.14) 13.58(0.03) 3.97(0.14)
2.5% 2.8% 0.36(0.55) 0.08(0.78) 0.44(0.80) 4.47(0.61) 2.13(0.34)
1.0% 1.2% 0.38(0.54) 2.45(0.12) 2.83(0.24) 21.14(0.00) 47.69(0.00)
QS 7.5% 8.6% 0.51(0.48) 0.06(0.80) 0.43(0.81) 8.21(0.22) 5.53(0.06)
5.0% 5.3% 0.18(0.67) 0.55(0.46) 0.75(0.69) 10.34(0.11) 5.39(0.07)
2.5% 2.1% 0.69(0.40) 0.86(0.35) 1.54(0.46) 8.57(0.19) 21.89(0.00)
1.0% 1.0% 0.00(1.00) 0.18(0.67) 0.18(0.91) 14.68(0.02) 95.41(0.00)
ES 7.5% 8.0% 0.35(0.55) 0.10(0.75) 0.34(0.84) 9.02(0.17) 5.26(0.07)
5.0% 5.0% 0.00(1.00) 0.92(0.34) 0.92(0.63) 8.89(0.17) 4.56(0.10)
2.5% 2.2% 0.38(0.53) 0.94(0.33) 1.32(0.51) 7.96(0.24) 23.53(0.00)
1.0% 0.9% 0.10(0.75) 0.14(0.70) 0.25(0.88) 13.84(0.03) 37.52(0.00)
RQS 7.5% 8.6% 1.67(0.19) 0.09(0.75) 1.50(0.47) 0.08(0.08) 3.93(0.14)
5.0% 5.5% 0.51(0.47) 0.36(0.55) 0.88(0.64) 12.46(0.05) 2.68(0.26)
2.5% 2.4% 0.04(0.84) 1.13(0.28) 1.17(0.55) 13.43(0.03) 12.78(0.00)
1.0% 0.7% 1.02(0.31) 0.08(0.28) 1.09(0.58) 1.56(0.95) 32.18(0.00)
RES 7.5% 7.9% 0.23(0.63) 0.00(0.97) 0.13(0.93) 9.02(0.17) 5.46(0.07)
5.0% 5.5% 0.51(0.47) 0.36(0.55) 0.88(0.64) 10.94(0.09) 1.44(0.46)
2.5% 2.4% 0.04(0.84) 1.13(0.29) 1.17(0.56) 7.53(0.27) 4.71(0.10)
1.0% 1.0% 0.00(1.00) 0.18(0.67) 0.18(0.91) 12.59(0.05) 38.64(0.00)
29
20

(see, Figure 2 and comments below for a discussion) in the correct direction such
that most of the empirical departures from the theoretical coverage are removed.
The empirical exception rates tend to stabilize around the nominal levels without
generating dependence or clustering in the exceptions. As a result, all covariateextended
CAViaR models are able to amply pass the three backtesting analyses of
Christo¤ersen (1988) at any of the usual con…dence levels. Similarly, the DQ test
tends to largely support the suitability of the extended models, showing sizeable
statistical gains with respect to the restricted case in the vast majority of cases
analyzed. Finally, and in sharp contrast to the results from returns-based methodologies,
the V QR test yields supportive evidence for correct conditional coverage
in the extended quantile regression setting, particularly, for the set of variables in
the volume group and for quantiles larger than 1%: For the 1% quantile, however,
the V QR rejects the null hypothesis of correct coverage. In view of the overall success,
particularly for variables such as Number of Trades, for which the remaining
testing procedures strongly support the correct coverage hypothesis. Under the
V QR metric, therefore, none of the di¤erent models and methodologies used in
our analysis are able to pass the back tests when addressing the 1% percentile. 10
Coupled with the evidence presented in the predictive-regression analysis, the
overall evidence in the out-of-sample results allows us to conclude that the degree
of liquidity and the trading conditions that characterize the market, as proxyed
by the di¤erent variables considered and, particularly, by those related to volume
characteristics, are predictors of the day-ahead conditional distribution of daily
returns and improve the out-of-sample performance of the VaR forecasts. For
extreme quantiles, such as 1%, the statistical evidence in our analysis is less conclusive:
while the testing procedures based on quantile-regression inference do not
support predictability for this percentile, the remaining back tests analyses do.
[Insert Figure 2 around here]
It is interesting to discuss in greater detail the di¤erences between the forecasts
from the restricted and unrestricted CAViaR models. To this end, Figure
2 displays the one-day 95% VaR forecasts from the restricted Restricted-CAViaR
model (solid line) against the unrestricted CAViaR model extended with either
Relative Quoted Spread (RQS) or Volume (black and grey dashed lines respectively).
As shown in Table 4, the actual proportion of VaR exceptions from the
Restricted-CAViaR model over the sample is much higher than the expected 5%,
and consequently the model is biased towards underestimating the actual level of
market risk in our sample. As depicted in Figure 2, the inclusion of the proxies for
market liquidity and trading activity generates an upward shift in the dynamics of
the predicted VaR process and introduces further variability in the forecasts with
respect to the predictions from the restricted model. As a result, the gap between
10 This may be symptomatic of size distortions. Piazza et al. (2009) analyze the small-sample
properties of the test through Monte Carlo simulation. Under experimental conditions, it is
shown that the VQR tends to reject the null model more frequently than expected, particularly,
for small quantiles such as = 0:01. On real data, the true characteristics of the unknown data
generating process may interfere with the asymptotic properties of the test and introduce further
distortions.
13
21

Figures
6
4
2
Market Portfolio Returns
0
­2
­4
­6
Out of Sample Period
­8
1988/01/04 1991/10/01 1995/06/27 1999/03/25 2002/12/31
Observations(Data)
Figure 1: Returns of the volume weighted market portfolio.
3.5
3
Restricted
Volume
RQS
ForecastsVaR (5% nominal level)
2.5
2
1.5
1
1998/09/10 1999/09/10 2000/09/10 2001/09/10
Observations (Data)
Figure 2: Forecasts VaR at nominal level 5% of Restricted-CAViaR
model (solid line) and Unrestricted CAViaR model extended with
volume and Relative Quoted Spread (RQS) variable (grey and black
dashed lines respectively).
34
22

the expected and the actual proportion of exceptions is reduced, which improves
the unconditional performance of the model, as discussed previously. Furthermore,
and as is revealed from the back test analysis, this improvement is achieved without
generating clusters of patches of exceptions. In view of overwhelming statistical
support given by all the backtesting procedures and the quantile predictive regression,
we therefore must conclude that the VaR predictions from the risk models
extended with the variables that proxy for liquidity and, particularly, volume, are
able to track the dynamics followed by the true VaR process more closely than a
purely-returns based model.
3.3 B/M and size portfolios
It is interesting to analyze the quantile-predictability of other representative portfolios.
Fama and French (1992) found that B/M and size characteristics seem to
capture most of the cross-section of average stock returns. Also, it is usual that
investors consider risk pro…les based on growth/value and size to make …nancial
decisions. Hence, it deserves interest to analyze predictability at di¤erent quantiles
for B/M-sorted (Low30 and High30) and Size-sorted (Low30 and High30)
portfolios. The results based on predictive quantile regressions are similar to those
discussed previously. We therefore discuss the main results for the out-of-sample
analysis, since some meaningful di¤erences worthy of comment arise in this case.
[Insert Tables 6.1 and 6.2 around here]
Tables 6.1 and 6.2 report the main outcomes from the backtesting analysis of
the CAViaR models extended with trade- and order-related variables, respectively,
for the Low30 and High30 B/M portfolios given 2 11 . There are meaningful
di¤erences across these portfolios. The LR UC and LR CC tests tend to indicate
a correct performance for all the covariate-extended models for both portfolios.
However, the DQ and, particularly, the V QR test, show a much more conservative
picture about the overall predictability of the Low30 portfolio, particularly, for
spread-related variables. In contrast, the overall evidence of predictability for the
High30 B/M portfolio is much stronger. All the back tests, including V QR, suggest
that the spread-related variables (particularly, e¤ective and relative e¤ective
spread) tend to predict correctly the distribution of the return at the quantiles
analyzed. Therefore, the extent of empirical predictability varies attending to
growth/value pro…les of the portfolios involved, which in turn favours certain variables
over others. As in the case of the market portfolio, the V QR test rejects
the correct performance of all the returns-based risk models for all the quantiles
2 involved, both in the Low30 and High30 portfolios (results not reported
for saving space). Therefore, the overall evidence suggests that models including
state variables that capture di¤erent aspects of the trading process outperform
models based on returns exclusively.
[Insert Tables 7.1 and 7.2 around here]
11 The results of the independence test LR IND are not presented in this section in order to
save space, although these results are available upon request.
14
23

Table 6.1: Backtesting VaR analysis for volume extended CAViaR models. B/M portfolio (Low 30 and High30). See details in
Table 4.
B/M (Low30)
X t Exc. LRUC LRCC DQ VQR
V 7.5% 8.7% 1.98(0.15) 1.72(0.42) 11.92(0.06) 2.09(0.35)
5.0% 5.5% 0.51(0.47) 0.52(0.77) 8.05(0.23) 2.11(0.35)
2.5% 2.3% 0.16(0.68) 0.56(0.75) 5.44(0.49) 6.09(0.05)
1.0% 1.0% 0.00(1.00) 3.17(0.20) 14.32(0.03) 15.30(0.00)
NT 7.5% 8.8% 2.31(0.12) 2.06(0.35) 11.92(0.06) 2.09(0.35)
5.0% 5.9% 1.61(0.20) 1.69(0.42) 15.57(0.02) 2.38(0.30)
2.5% 2.5% 0.04(0.83) 2.47(0.28) 7.98(0.24) 3.71(0.16)
1.0% 1.0% 0.00(1.00) 3.17(0.20) 12.86(0.05) 23.05(0.00)
NS 7.5% 8.6% 1.67(0.19) 1.41(0.49) 14.61(0.02) 2.06(0.36)
5.0% 5.4% 0.33(0.56) 0.34(0.84) 8.01(0.24) 1.23(0.54)
2.5% 2.4% 0.04(0.83) 2.47(0.28) 8.27(0.22) 2.30(0.30)
1.0% 0.7% 1.01(0.31) 5.73(0.05) 19.33(0.00) 45.13(0.00)
NSS 7.5% 8.4% 1.12(0.28) 0.91(0.63) 11.36(0.08) 2.51(0.29)
5.0% 5.6% 0.73(0.39) 0.74(0.68) 10.82(0.09) 2.20(0.33)
2.5% 2.2% 0.38(0.53) 0.87(0.64) 4.74(0.58) 4.30(0.12)
1.0% 0.9% 0.10(0.74) 3.72(0.05) 15.46(0.02) 14.45(0.00)
TVD 7.5% 8.9% 2.67(0.10) 2.43(0.29) 14.04(0.03) 3.40(0.18)
5.0% 6.4% 3.80(0.05) 4.08(0.12) 15.60(0.02) 3.20(0.20)
2.5% 2.7% 0.16(0.68) 1.88(0.38) 6.42(0.38) 5.46(0.07)
1.0% 1.4% 1.43(0.23) 3.31(0.19) 11.66(0.07) 26.72(0.00)
B/M (High30)
Exc. LRUC LRCC DQ VQR
8.8% 2.31(0.12) 2.86(0.23) 4.36(0.63) 3.37(0.19)
6.9% 6.83(0.01) 8.02(0.01) 12.09(0.06) 12.83(0.00)
3.6% 4.37(0.04) 4.46(0.10) 14.77(0.02) 9.87(0.01)
1.3% 0.83(0.36) 1.17(0.55) 9.05(0.17) 4.51(0.10)
8.9% 2.68(0.10) 3.07(0.21) 4.10(0.66) 4.99(0.08)
6.8% 6.17(0.01) 7.51(0.02) 11.96(0.06) 13.26(0.00)
3.1% 1.38(0.24) 1.39(0.49) 7.77(0.25) 1.82(0.40)
1.3% 0.83(0.36) 2.83(0.24) 19.17(0.00) 0.38(0.83)
8.8% 2.32(0.13) 2.86(0.23) 4.16(0.65) 3.85(0.15)
6.8% 6.16(0.01) 7.51(0.02) 12.08(0.06) 12.74(0.00)
3.2% 1.85(0.17) 1.86(0.39) 7.81(0.25) 11.13(0.00)
1.2% 0.38(0.54) 2.67(0.26) 18.49(0.01) 0.38(0.83)
8.9% 2.67(0.10) 3.07(0.21) 4.12(0.66) 3.73(0.15)
6.7% 5.52(0.02) 6.14(0.04) 8.76(0.19) 15.51(0.00)
3.2% 1.84(0.19) 1.86(0.39) 7.70(0.26) 10.47(0.01)
1.5% 2.18(0.13) 2.62(0.26) 17.72(0.00) 15.79(0.00)
9.0% 3.06(0.08) 3.31(0.19) 4.21(0.65) 4.09(0.13)
6.8% 6.16(0.01) 7.51(0.02) 12.33(0.05) 11.12(0.00)
3.4% 2.99(0.08) 3.61(0.16) 10.19(0.12) 6.80(0.03)
1.0% 0.00(1.00) 0.18(0.91) 0.43(0.99) 0.06(0.97)
30
24

Table 6.2: Backtesting VaR analysis for spread extended CAViaR models. B/M portfolio (Low 30 and High30). See details in
Table 4.
B/M (Low30)
X t Exc. LRUC LRCC DQ VQR
QS 7.5% 8.9% 2.67(0.10) 2.43(0.29) 14.19(0.03) 4.89(0.09)
5.0% 5.5% 0.51(0.47) 0.52(0.77) 16.95(0.01) 12.49(0.00)
2.5% 2.1% 0.69(0.40) 1.58(0.45) 12.10(0.06) 80.92(0.00)
1.0% 0.7% 1.01(0.31) 1.09(0.57) 3.29(0.77) 41.48(0.00)
ES 7.5% 8.4% 1.12(0.28) 0.91(0.63) 15.74(0.02) 5.34(0.07)
5.0% 5.4% 0.32(0.56) 0.34(0.84) 0.05(0.05) 10.63(0.00)
2.5% 2.5% 0.00(1.00) 0.23(0.88) 5.55(0.05) 29.39(0.00)
1.0% 0.9% 0.10(0.74) 0.24(0.88) 2.38(0.88) 22.26(0.00)
RQS 7.5% 9.1% 3.47(0.06) 3.28(0.19) 15.45(0.01) 5.31(0.07)
5.0% 5.3% 0.18(0.66) 0.74(0.68) 14.77(0.02) 22.93(0.00)
2.5% 2.5% 0.00(1.00) 0.23(0.88) 9.81(0.13) 55.98(0.00)
1.0% 1.0% 0.00(1.00) 0.18(0.91) 2.23(0.89) 12.81(0.00)
RES 7.5% 8.4% 1.12(0.28) 0.91(0.63) 10.99(0.08) 2.91(0.23)
5.0% 5.6% 0.73(0.39) 1.02(0.59) 14.19(0.02) 22.63(0.00)
2.5% 2.5% 0.00(1.00) 0.23(0.88) 9.20(0.16) 14.33(0.00)
1.0% 1.4% 1.43(0.23) 3.31(0.19) 18.04(0.01) 45.92(0.00)
B/M (High30)
Exc. LRUC LRCC DQ VQR
8.6% 1.67(0.19) 1.89(0.38) 2.69(0.84) 3.54(0.17)
6.1% 2.38(0.12) 2.91(0.23) 7.51(0.27) 4.89(0.09)
2.8% 0.35(0.55) 0.43(0.80) 17.05(0.01) 1.11(0.57)
0.8% 0.43(0.51) 0.55(0.75) 12.28(0.06) 0.14(0.93)
8.2% 0.68(0.40) 1.49(0.47) 2.83(0.83) 2.56(0.28)
5.7% 0.98(0.32) 1.99(0.36) 6.58(0.36) 3.92(0.14)
2.4% 0.04(0.83) 0.34(0.83) 4.76(0.57) 2.62(0.27)
0.9% 0.10(0.74) 0.26(0.87) 20.00(0.00) 0.18(0.91)
8.9% 2.67(0.10) 3.07(0.21) 4.40(0.62) 5.45(0.07)
6.3% 3.29(0.06) 3.65(0.16) 7.16(0.31) 7.67(0.02)
2.9% 0.62(0.42) 0.67(0.71) 16.10(0.01) 3.41(0.18)
1.0% 0.00(1.00) 0.18(0.91) 21.14(0.00) 6.25(0.04)
8.8% 2.31(0.12) 2.87(0.23) 4.09(0.66) 3.67(0.16)
6.3% 3.29(0.06) 4.44(0.11) 7.06(0.32) 8.77(0.01)
2.4% 0.04(0.83) 0.34(0.84) 11.77(0.07) 4.37(0.11)
1.3% 0.83(0.36) 1.15(0.56) 19.21(0.00) 2.53(0.28)
31
25

Table 7.1: Backtesting VaR analysis for volume extended CAViaR models. Size portfolio (Low30 and High30). See details in
Table 4.
SIZE (Low30)
X t Exc. LRUC LRCC DQ VQR
V 7.5% 7.2% 0.13(0.71) 0.29(0.86) 3.77(0.71) 1.41(0.49)
5.0% 3.8% 3.29(0.06) 3.50(0.17) 8.75(0.19) 1.90(0.39)
2.5% 2.3% 0.16(0.68) 1.24(0.53) 2.00(0.92) 1.90(0.39)
1.0% 1.4% 1.43(0.23) 1.84(0.39) 2.84(0.83) 6.13(0.05)
NT 7.5% 7.4% 0.50(0.47) 0.51(0.77) 7.10(7.11) 2.65(0.27)
5.0% 3.9% 2.74(0.09) 2.90(0.23) 7.79(0.25) 2.75(0.25)
2.5% 2.20% 0.38(0.53) 1.36(0.50) 2.25(0.89) 6.01(0.05)
1.0% 1.2% 0.37(0.53) 0.67(0.71) 1.19(0.98) 3.99(0.14)
NS 7.5% 7.3% 0.05(0.80) 0.17(0.91) 4.25(0.64) 4.72(0.09)
5.0% 3.8% 3.29(0.06) 3.50(0.17) 8.72(0.19) 2.80(0.25)
2.5% 2.2% 0.38(0.53) 1.36(0.50) 2.26(0.90) 2.80(0.25)
1.0% 1.2% 0.37(0.53) 0.67(0.71) 1.16(0.98) 7.95(0.02)
NSS 7.5% 7.3% 0.05(0.80) 0.17(0.91) 3.36(0.76) 5.89(0.05)
5.0% 3.8% 3.29(0.06) 0.62(0.17) 8.75(0.19) 3.26(0.20)
2.5% 2.1% 0.69(0.40) 1.58(0.45) 2.68(0.85) 6.16(0.05)
1.0% 1.3% 1.17(0.55) 1.17(0.55) 1.93(0.92) 1.92(0.38)
TVD 7.5% 7.5% 0.00(1.00) 0.04(0.97) 5.15(0.52) 5.10(0.08)
5.0% 3.6% 4.55(0.03) 4.92(0.08) 9.62(0.14) 2.29(0.32)
2.5% 2.3% 0.16(0.68) 1.24(0.53) 2.00(0.92) 7.17(0.03)
1.0% 1.2% 0.37(0.53) 0.67(0.71) 1.16(0.98) 1.01(0.60)
SIZE (High30)
Exc. LRUC LRCC DQ VQR
8.7% 1.98(0.15) 1.72(0.42) 9.98(0.13) 2.17(0.34)
5.9% 1.61(0.20) 2.09(0.35) 9.99(0.12) 2.50(0.29)
2.6% 0.04(0.84) 1.98(0.37) 3.94(0.68) 1.28(0.53)
0.7% 1.01(0.31) 1.09(0.57) 2.80(0.83) 11.96(0.00)
8.7% 1.98(0.15) 1.72(0.42) 9.54(0.15) 1.56(0.46)
6.0% 1.98(0.15) 1.71(0.42) 8.53(0.20) 2.56(0.28)
2.6% 0.04(0.84) 1.98(0.37) 4.76(0.57) 0.78(0.68)
0.7% 1.01(0.31) 1.09(0.57) 2.71(0.84) 14.94(0.00)
8.8% 2.31(0.12) 2.06(0.35) 9.59(0.14) 1.90(0.39)
5.5% 0.51(0.47) 1.73(0.42) 8.99(0.17) 2.50(0.29)
2.4% 0.04(0.83) 0.34(0.83) 2.66(0.85) 2.03(0.36)
0.6% 1.88(0.16) 1.93(0.37) 2.48(0.87) 13.09(0.00)
8.6% 1.67(0.19) 1.41(0.49) 9.31(0.16) 1.32(0.52)
5.8% 1.28(0.25) 1.92(0.38) 9.94(0.13) 1.74(0.42)
2.1% 0.69(0.40) 1.54(0.46) 3.67(0.72) 1.15(0.57)
0.6% 1.88(0.16) 1.93(0.37) 2.63(0.85) 15.20(0.00)
9.2% 3.90(0.04) 3.57(0.16) 12.04(0.06) 3.37(0.19)
6.3% 3.29(0.06) 4.04(0.13) 12.25(0.06) 3.56(0.17)
3.3% 2.38(0.12) 3.12(0.20) 6.53(0.37) 2.22(0.33)
1.0% 0.00(1.00) 3.17(0.20) 12.47(0.05) 10.82(0.00)
32
26

Table 7.2: Backtesting VaR analysis for spreads extended CAViaR models. Size portfolio (Low30 and High30). See details in
Table 4.
SIZE (Low30)
X t Exc. LRUC LRCC DQ VQR
QS 7.5% 7.6% 0.01(0.90) 0.03(0.98) 8.14(0.23) 3.31(0.19)
5.0% 4.1% 1.81(0.17) 1.85(0.39) 10.36(0.11) 8.05(0.02)
2.5% 2.4% 0.04(0.83) 1.22(0.54) 5.43(0.49) 7.06(0.03)
1.0% 1.5% 2.18(0.13) 2.65(0.26) 4.18(0.65) 3.01(0.22)
ES 7.5% 7.2% 0.13(0.71) 0.29(0.86) 9.51(0.15) 3.37(0.19)
5.0% 3.8% 3.29(0.06) 3.47(0.17) 9.61(0.14) 3.34(0.19)
2.5% 2.3% 0.16(0.68) 1.24(0.53) 5.81(0.44) 9.02(0.01)
1.0% 1.5% 2.18(0.13) 2.65(0.26) 4.36(0.63) 2.14(0.34)
RQS 7.5% 7.6% 0.01(0.90) 0.34(0.84) 10.27(0.11) 2.58(0.27)
5.0% 4.1% 1.81(0.17) 1.85(0.39) 9.45(0.15) 11.62(0.00)
2.5% 2.3% 0.16(0.68) 1.24(0.53) 2.03(0.92) 4.21(0.12)
1.0% 1.4% 1.43(0.23) 1.84(0.39) 7.30(0.29) 2.62(0.27)
RES 7.5% 7.3% 0.05(0.80) 0.17(0.91) 7.55(0.27) 3.46(0.18)
5.0% 3.8% 3.29(0.06) 3.47(0.17) 8.84(0.18) 9.69(0.01)
2.5% 2.1% 0.69(0.40) 1.58(0.45) 2.66(0.85) 15.37(0.00)
1.0% 1.4% 1.43(0.23) 1.84(0.39) 2.89(0.82) 2.21(0.33)
SIZE (High30)
Exc. LRUC LRCC DQ VQR
8.4% 1.12(0.28) 0.91(0.63) 13.92(0.03) 6.45(0.04)
5.7% 0.98(0.32) 3.12(0.21) 15.12(0.02) 4.52(0.10)
2.1% 0.69(0.40) 1.54(0.46) 4.91(0.56) 7.87(0.02)
0.6% 1.88(0.16) 1.95(0.37) 3.47(0.75) 76.58(0.00)
8.2% 0.68(0.40) 0.55(0.75) 12.67(0.05) 6.45(0.04)
5.4% 0.32(0.56) 1.77(0.41) 12.76(0.05) 4.12(0.13)
2.1% 0.69(0.40) 1.29(0.52) 4.22(0.65) 10.49(0.01)
0.7% 0.00(1.00) 1.10(0.57) 4.90(0.56) 12.61(0.00)
8.5% 1.38(0.23) 1.14(0.56) 11.38(0.08) 2.58(0.27)
5.8% 1.28(0.25) 1.92(0.38) 17.27(0.01) 4.23(0.12)
2.5% 0.00(1.00) 1.23(0.54) 4.13(0.66) 2.28(0.32)
0.8% 0.43(0.51) 0.54(0.76) 1.86(0.93) 13.29(0.00)
6.9% 0.53(0.46) 0.90(0.63) 11.31(0.08) 2.36(0.31)
5.7% 0.98(0.32) 0.98(0.60) 14.84(0.02) 8.76(0.01)
2.2% 0.38(0.38) 0.87(0.64) 5.28(0.51) 3.52(0.17)
0.5% 3.09(0.07) 3.12(0.20) 2.97(0.81) 70.59(0.00)
33
27

Finally, Tables 7.1 and 7.2 show the main outcomes for the backtesting analysis
on the Low30 and High30 Size-sorted portfolios given the variables in the volume
and liquidity groups, respectively. The results in this analysis are, in general, more
similar to those discussed for the volume-weighted market portfolio. The backtesting
analysis reveals a great performance of both volume- and liquidity-extended
risk models to forecast the VaR of the Low30 portfolio at any of the quantiles. The
best results are observed for volume-related variable, for which all the back tests,
including the V QR, tend to accept the suitability of the model. For the High30
portfolio, however, the results are more conservative and in line with those already
discussed for the volume-weighted portfolio: volume and liquidity proxies seem
to make a good job for most quantiles under the di¤erent tests, with the overall
evidence being more ambiguous for the 1% percentile in the VQR test. This suggests
that the conditional distribution of small capitalization stocks is predictable,
particularly, when including volume-related variables. It should be noted that the
overall performance of returns-based procedure (not presented) seems better for
small capitalization assets, with the extreme value theory models yielding the best
performance throughout. Nevertheless, the performance is inferior to the extended
models and, as in the previous cases, none of the returns-based models are able to
pass the V QR test.
4 Concluding remarks
In this paper, we have analyzed the predictability of the tail of the conditional
distribution of market returns using di¤erent variables that are widely related to
trading activity and market liquidity. Our methodological approach has mainly
built on the quantile regression methodology and, particularly, the CAViaR quantile
regression setting proposed in Engle and Manganelli (2004). This strategy
allows us to study in a simple and direct way the forecasting ability of a number
of covariate-extended models in relation to a restricted model that relies solely on
returns, using both predictive regressions and backtesting analysis.
The extent of statistical evidence supporting predictability may vary depending
mainly on the size of the target quantile. However, the overall evidence strongly
suggests the existence of predictability of the conditional distribution on the basis
of trading activity and liquidity variables. This evidence is particularly strong
for the set of volume-related variables in the market portfolio and, more generally,
is fairly robust against the consideration of di¤erent representative market
portfolios, di¤erent proxy variables of liquidity and trading activity, and di¤erent
testing procedures. Consequently, the main conclusion from this paper is that the
day-ahead conditional distribution of returns is predictable when using observable
market information which is not necessarily limited to returns and, therefore, such
information can be used in downside risk modelling.
Finally, although our analysis has focused on the analysis of quantiles and,
therefore, allows us to obtain direct conclusions for the VaR methodology, the
main conclusions from our analysis may be extrapolated to other quantile-based
downside risk measures, among which expected shortfall (conditional VaR) is the
most representative. If quantiles are predictable, trivial transformations such as
15
28

the mean of quantiles should be predictable as well. The formal analysis of this
interesting issue is left for future research.
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29

Appendix A: VaR models
In Section 4 we compare the performance of CAViaR model with other standard
procedures to compute VaR. These include the EWMA, GARCH and Extreme
Value Theory methods. The common setting in these parametric models assumes
that (conditionally demeaned) returns obey dynamics given by
r t = t t ; t jF t 1 iidN (0; 1) (A.1)
where t denotes the conditional volatility of the process. We brie‡y discuss the
main settings of these approaches in the sequel.
A. VaR EWMA
RiskMetrics popularized the Exponential Weighting Moving Average (EWMA)
scheme as an easy way to model the volatility process. The latent volatility dynamics
are assumed to obey the recursive dynamics:
b 2 t = b 2 t 1 + (1 )r 2 t 1; t = 1; :::; T (A.2)
with the initial condition b 2 0 = r0 2 = E (rt 2 ) : The smoothing parameter 0 1
can be estimated, although RiskMetrics advises the setting = 0:95 for data
recorded on a daily basis. Then, the one-day ahead forecast given F T is simply
given by b 2 T +1jT = b 2 T + (1 )rT 2 :
RiskMetrics further assumes the particularly strong assumption that the innovations
t are conditionally normal distributed, from which the one-period ahead
VaR forecast would be given by Z b T +1jT ; with Z denoting the -quantile of
the standard normal distribution. In order to ensure robustness against departures
from normality, we proceed in a slightly di¤erent way. Let b t = r t =b t be
the estimated innovations given the estimates of the EWMA volatility process,
and let Q (b t ) be the unconditional -quantile of the empirical distribution of b t .
Then, a ‘robusti…ed’VaR forecast that does not rely upon distributional assumption
is given by:
V aR ;t+1 (EW MA) = Q (b t )b T +1jT (A.3)
B. VaR GARCH
The simplest GARCH (1,1) model is the most popular approach to model and
forecast market risk in practice due to its impressive performance and statistical
properties (Hansen and Lunde, 2005). The standard GARCH(1,1) model assumes
that daily returns obey dynamics given by:
r t = t t ; t jF t 1 iidN (0; 1) (A.4)
2 t = ! + " 2 t 1 + 2 t 1
with the restrictions ! > 0; ; 0 ensuring that the conditional variance
process is well-de…ned. Although …nancial returns are known to be non-normally
distributed, the Gaussian assumption is particularly convenient because it ensures
parameter consistency under certain regularity conditions even in the absence of
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30

normality. Parameters can thus be estimated by (quasi) maximum likelihood estimation,
yielding a consistent estimate of the conditional variance process. The
day-ahead forecast is then given by:
b 2 T +1jT = b! + br 2 T + b b 2 T
(A.5)
As in the EWMA model discussed previously, given the GARCH estimates b t
and the resultant standardized innovations, b t = r t =b t ; the ‘robust’one-day VaR-
GARCH forecast is determined as:
V aR ;T +1 (GARCH) = Q (b t )b T +1jT : (A.6)
with Q (b t ) denoting the -quantile of the empirical distribution.
C. VaR Extreme Value Theory
This method can be seen as a parametric re…nement of the previous approaches.
Essentially, the procedure requires the characterization of the tail behavior of the
set of i.i.d. innovations t in the return process. To circumvent the problem that t
is not observable directly, the estimated residuals b t = r t =b t can be used instead,
with b t determined according to some volatility model, such as those discussed
previously. Since GARCH estimates tend to outperform any other procedure, we
estimate the empirical process b t on the basis of the GARCH(1,1) estimates as
discussed above.
The rest of the procedure is described as follows. Given the series b t ; the total
sample period is divided into B = 740 blocks of length l = 5 observations to record
the maximum value of each block (i.e., the maximum loss in the period), say m b ,
b = 1; :::; B; in a time-series process. The Extreme Value Theory suggests …tting
the Generalized Extreme Value distribution (GEV, also known as Fisher–Tippett
distribution) to this series. The GEV arises as the limit distribution of properly
normalized maxima of a sequence of i.i.d. random variables, and is characterized
by the density function
f (z b ; 1 ; 2 ; 3 ) =
1 1=3 1
n
[1 +
3 z b ] exp
2
[1 + 3 z b ]
1= 3
o
(A.7)
if z b > 1; where z b = (m b 1 ) = 2 denotes the standardized variable. The
(unknown) parameters characterize the shape ( 1 ), scale ( 2 ) and location ( 3 ) of
the distribution and can be estimated consistently by di¤erent methods, such as
maximum-likelihood. The importance of this approach is that by inverting this
distribution (with the unknown parameters replaced by their consistent estimates),
we can go from the asymptotic GEV distribution of maxima to the distribution of
the observations themselves and obtain a closed-form expression for the unconditional
VaR of b t given ; namely,
"
b
Q (b t ) = b 2
3 1
b 1
log 1
# b1
1
l
(A.8)
Finally, as in the EWMA and GARCH approaches, we generate the one-day ahead
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31

VaR forecast as
with b T +1jT determined as in (A:5) :
V aR ;T +1 (EV T ) = b T +1jT Q (b t ) (A.9)
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32

Appendix B: Backtesting analysis
I) Unconditional test, LR UC :
The most basic assumption is that the market risk model provides a correct
unconditional coverage, namely, H 0;U : E [H ;t ] = . The null hypothesis is rejected
for large values of the likelihood-ratio test de…ned as
LR UC = 2(N N ) log(1
N
N ) log(1 ) + 2N log N
N
log 2 (1)
P
t=1;N
(B.1)
where 2 (1) stands for a Chi-squared distribution with one degree of freedom, N
H t; is the number of exceptions, and N is the total number of out-ofsample
observations. Note that N =N = b H is simply the sample mean of H ;t ,
i:e, the sample equivalent of E [H ;t ] :
II) Independence test, LR IND :
If exceptions are serially correlated, the property of reliability conditional coverage
will be defective even if the unconditional coverage is correct, because the
risk of bankruptcy is higher. Christo¤ersen (1998) proposes the analysis of the
…rst-order serial correlation in H ;t through a binary …rst-order Markov chain with
transition probability matrix
=
1 01 01
; with
1 11 ij = Pr(H ;t = j j H ;t 1 = i); i; j 2 f0; 1g (B.2)
01
The approximate joint likelihood conditional on the …rst observation is
L(; H ;t j H ;1 ) = (1 01 ) n 00
n 01
01 (1 11 ) n 10
n 11
11 ; (B.3)
where n ij represents the number of transitions from state i to state j. The
maximum-likelihood estimators under the alternative hypothesis are b 01 = n 01 = (n 00 + n 01 ) ;
and b 11 = n 11 = (n 10 + n 11 ) : Under the null hypothesis of independence, we have
01 = 11 = 0 ; with 0 = ; from which the conditional binomial joint likelihood
is
L( 0 ; H ;t j H ;1 ) = (1 01 ) n 00+n 10
n 01+n 11
01 : (B.4)
Note that 0 can be estimated as b 0 = N =N. The likelihood ratio test for the
hypothesis of independence is given by
h
i
LR IND = 2 log L(^; H ;t j H 1 ) log L(b 0 ; H ;t j H ;1 ) 2 (1)
(B.5)
III) Conditional test, LR CC :
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33

Finally, we can study simultaneously whether the VaR violations are independent
and occur with the correct probability, i.e., H 0;C : E [H ;t jF t 1 ] = . Because
b 0 is unconstrained, the test in equation (B.5) does not impose the correct coverage.
Christo¤ersen (1998) devised a joint test for independence and correct
coverage (i.e., correct conditional coverage) by combining the previous tests:
h
i
LR CC = 2 log L(^; H ;T j H 1 ) log L(; H T j H 1 ) 2 (2) (B.6)
This is equivalent to testing if the sequence of H ;t is independent and the probabilities
to observe an exception given the set of information is equal to the nominal
level , namely, 01 = 11 = . Therefore, we can write
LR CC = LR UC + LR IND ;
(B.7)
which provides the suitable test statistic to check whether H ;t exhibits correct
conditional coverage properties. Since the test involves two restrictions, the asymptotic
convergence to a 2 (2) distribution.
IV) Dynamic Quantile test, DQ (Engle and Manganelli, 2004).
The LR CC test does not have the power to detect higher-order dependences
in H ;t . Engle and Manganelli (2004) introduced a test that accounts for a more
general form of dependence in order to test H 0;C : E [H ;t jF t 1 ] = . De…ne the
time series:
Hit t = H ;t (B.8)
and let the matrix of instrumental variables
Z t = [Hit t 1 ; Hit t 2 ; :::; Hit t p ; V aR ;t 1 ; :::; V aR ;t q ] : (B.9)
for some predetermined, …xed lag values p; q 1. Note that H 0;C : E [H ;t jF t 1 ] =
implies the martingale condition E [Hit ;t jF t 1 ] = 0, which in turn could tested
as E [Hit ;t jZ t ] = 0 for suitable values of p and q. Following Engle and Manganelli
(2004), we set p = 4; q = 1 and test the martingale restriction through ordinary
least-squares analysis in the auxiliary regression Hit t = Z t + u t by analyzing the
joint restriction H 0 : = 0 through a standard F test, given by the test statistic
DQ = b 0 Z 0 tZ t
b
(1 )
(B.10)
with b denoting the OLS estimate of : Under the null hypothesis, DQ 2 (p+q) .
V) VaR Quantile Regression test, V QR (Piazza et al., 2009).
Given the VaR forecasts V aR ;t , t = 1; :::; N; Piazza et al. (2009) focus on the
encompassing quantile regression
r t = 0; + 1; V aR ;t + " t; ; t = 1; ::; N
(B.11)
where r t denotes the realized returns over the out-of-sample period. The hypothesis
that V aR ;t is an optimal forecasts of the conditional -quantile of r t can be tested
21
34

through the joint restriction H 0 : 0; = 0; 1; = 1 or, equivalently, H 0 : = 0
with = ( 0 ; 1 1) 0 in the previous regression model.
Let b the quantile regression estimate of : Under standard regularity conditions,
the asymptotic distribution of p
N b is a normal with zero mean
and …nite variance . Under the null, it follows that
V QR = T [ b 0 1 b ] 2 (2)
where the covariance matrix can be estimated by usual methods.
(B.12)
References
[1] Adrian, T. and Brunnermeir, M.K. 2009. “CoVaR.” Working Paper, New
York.
[2] Alexander, C. and Sheedy, E. 2008. “Developing a Stress Testing Framework
Based on Market Risk Models.” Journal of Banking and Finance,
32(10):2220-2236.
[3] Bao, Y., Lee, T. and Salto B. 2006. “Evaluating Predictive Performance of
Value-at-Risk Models in Emerging Markets: A Reality Check.” Journal of
Forecasting, 25(2):101-128.
[4] Bassett, G.W. and Koenker, R. 1982. “An Empirical Quantile Function for
Linear Models with iid Errors.”Journal of the American Statistical Association,
77(378):407-415.
[5] Bollerslev, T. and Melvin, M. 1994. “Bid-Ask Spreads and the Volatility in the
Foreign Exchange Market: An Empirical Analysis.”Journal of International
Economics, 36(3-4):355-372.
[6] Cenesizoglu, T. and Timmerman, A. 2008. “Is the Distribution of Stocks
Returns Predictable.”Working Paper, University of California at San Diego.
[7] Chernozhukov, V. 2005. “Extremal Quantile Regression.”The Annals of Statistics,
33(2):806-839.
[8] Chordia, T., Roll, R. and Subrahmanyam, A. 2001. “Market Liquidity and
Trading Activity.”Journal of Finance, 56(2):501-530.
[9] Christo¤ersen, P.F. 1998. “Evaluating Interval Forecasts.”International Economic
Review, 39(4):841-862.
[10] Clark, P.K. 1973. “A Subordinated Stochastic Process Model with Finite Variance
for Speculative Prices.”Econometrica, 41(1):135-156.
[11] Cochrane, J.H. 2005. Asset Pricing (Revised Edition). Princeton University
Press: Princeton and Oxford.
22
35

[12] Fama, E.F. and French, K.R. 1992. “The Cross-Section of Expected Stock
Returns.”Journal of Finance, 47(8):427-465.
[13] Engle, R. and Manganelli, S. 2004. “CAViaR: Conditional Autoregressive
Value at Risk by Regression Quantiles.” Journal of Business and Economic
Statistics, 22(4):367-381.
[14] Go¤e, W.L., Ferrier, G.D. and Rogers, J. 1994. “Global Optimization of Statistical
Functions with Simulated Annealing.”Journal of Econometrics, 60(1-
2):65-99.
[15] Hansen, P.R. and Lunde, A. 2005. “A Forecast Comparison of Volatility Models:
Does Anything Beat a GARCH (1,1)?”Journal of Applied Econometrics,
20(7):873-889.
[16] Kalimipalli, M. and Warga, A. 2002. “Bid/Ask Spread, Volatility and Volume
in the Corporate Bond Market.”The Journal of Fixed Income, 11(4):31-42.
[17] Koenker, R. and Bassett, G. 1978. “Regression Quantiles.” Econometrica,
46(1):33-50.
[18] Kouretas, G.P. and Zarangas, L. 2005. “Conditional Ine¢ ciency Value at Risk
by Regression Quantiles: Estimating Market Risk for Major Stock Markets.”
Working Paper, Preveza, Greece.
[19] Kuester, K., Mittnik, S. and. Paolella, M.S. 2006. “Value-at-Risk Prediction:
A Comparison of Alternative Strategies.”Journal of Financial Econometrics,
4(1):53-89.
[20] Manganelli, S. and Engle,.R.F. 2004. A Comparison of Value-at-Risk Models
in Finance. Wiley: Chichester.
[21] McNeil, A.J., Frey, R. and Embrechts, P. 2005. Quantitative Risk Management:
Concepts, Techniques, and Tools. Princeton University Press: Princeton.
[22] Piazza, W., Renato, L., Linton, O. and Smith, D. 2009. “Evaluating Valueat-Risk
Models via Quantile Regression.” Working Paper. Forthcoming in
Journal of Business and Economic Statistics.
[23] Rapach, D.E. and Strauss, J.K. 2009. “Di¤erences in Housing Price Forecastability
Across U.S. States.” International Journal of Forecasting, 25(2):351-
372.
[24] Stoll, H.S. 1989. “Inferring the Components of the Bid-Ask Spread: Theory
and Empirical Tests.”Journal of Finance, 44(1):115-134.
[25] Suominen, M. 2001. “Trading Volume and Information Revelation in Stock
Markets.”The Journal of Financial and Quantitative Analysis, 36(4):545-565.
[26] Tauchen, G.E. and Pitts, M. 1983. “The Price Variability-Volume Relationship
on Speculative Markets.”Econometrica, 51(2):485-505.
23
36

[27] Taylor, J.W. 1999. “A Quantile Regression Approach to Estimating the Distribution
of Multiperiod Returns.”Journal of Derivatives, 7(1):64-78.
[28] Taylor, J.W. 2000. “A Quantile Regression Neural Network Approach to Estimating
the Conditional Density of Multiperiod Returns.” Journal of Forecasting,
19(4):299-311.
[29] Taylor, J.W. 2008. “Using Exponentially Weighted Quantile Regression to
Estimate Value at Risk and Expected Shortfall.”Journal of Financial Econometrics,
6(3):382-406.
[30] White, H. 1994. Estimation, Inference and Speci…cation Analysis. Cambridge
University Press: Cambridge.
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ad
serie
Ivie
Guardia Civil, 22 - Esc. 2, 1º
46020 Valencia - Spain
Phone: +34 963 190 050
Fax: +34 963 190 055
Department of Economics
University of Alicante
Campus San Vicente del Raspeig
03071 Alicante - Spain
Phone: +34 965 903 563
Fax: +34 965 903 898
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E-mail: publicaciones@ivie.es